Best Known (35, 35+32, s)-Nets in Base 128
(35, 35+32, 1024)-Net over F128 — Constructive and digital
Digital (35, 67, 1024)-net over F128, using
- 1282 times duplication [i] based on digital (33, 65, 1024)-net over F128, using
- t-expansion [i] based on digital (32, 65, 1024)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 1024, F128, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using
- net defined by OOA [i] based on linear OOA(12865, 1024, F128, 33, 33) (dual of [(1024, 33), 33727, 34]-NRT-code), using
- t-expansion [i] based on digital (32, 65, 1024)-net over F128, using
(35, 35+32, 5466)-Net over F128 — Digital
Digital (35, 67, 5466)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12867, 5466, F128, 3, 32) (dual of [(5466, 3), 16331, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12867, 16398, F128, 32) (dual of [16398, 16331, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(26) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(31) ⊂ Ce(26) [i] based on
- OOA 3-folding [i] based on linear OA(12867, 16398, F128, 32) (dual of [16398, 16331, 33]-code), using
(35, 35+32, large)-Net in Base 128 — Upper bound on s
There is no (35, 67, large)-net in base 128, because
- 30 times m-reduction [i] would yield (35, 37, large)-net in base 128, but